### Table 14. Therefore the normal form of the D4-equivariant map f in (1), with IRq = V is

"... In PAGE 31: ...= [x1; x2; 0; 0]; b2 = [0; 0; x1; x2]; b3 = [x3; x4; 0; 0]; b4 = [0; 0; x3; x4]; b5 = [x13; x23; 0; 0]; b6 = [0; 0; x13; x23]; b7 = [x12x3; x22x4; 0; 0]; b8 = [0; 0; x12x3; x22x4]; b9 = [x1x32; x2x42; 0; 0]; b10 = [0; 0; x1x32; x2x42]; b11 = [x33; x43; 0; 0]; b12 = [0; 0; x33; x43]; b13 = [x1x2x4; x1x2x3; 0; 0]; b14 = [0; 0; x1x2x4; x1x2x3]; b15 = [x2x4x3; x1x3x4; 0; 0]; b16 = [0; 0; x2x4x3; x1x3x4]; b17 = [x23x1x4; x13x2x3; 0; 0]; b18 = [0; 0; x23x1x4; x13x2x3]; b19 = [x13x32; x23x42; 0; 0]; b20 = [0; 0; x13x32; x23x42]; b21 = [x23x4x3; x13x3x4; 0; 0]; b22 = [0; 0; x23x4x3; x13x3x4]; b23 = [x12x33; x22x43; 0; 0]; b24 = [0; 0; x12x33; x22x43]; b25 = [x2x1x43; x1x2x33; 0; 0]; b26 = [0; 0; x2x1x43; x1x2x33]; b27 = [x2x43x3; x1x33x4; 0; 0]; b28 = [0; 0; x2x43x3; x1x33x4]; b29 = [x23x1x43; x13x2x33; 0; 0]; b30 = [0; 0; x23x1x43; x13x2x33]; b31 = [x23x43x3; x13x33x4; 0; 0]; b32 = [0; 0; x23x43x3; x13x33x4] Table14 : Generating set for the equivariants when D4 acts as #5 + #5. The parameters will be introduced later in order to unfold the linear part of (28); in which case at most B1; :::; B4 will depend on them.... ..."

### Table 11: Generating set for the equivariants when D4 acts as #3 + #5.

"... In PAGE 26: ... ii) A Hilbert basis for the ring of smooth IR-valued D4-invariant functions on V is given by i, i = 1; :::; 4, in Table 10. iii) The generators for the module of D4-equivariant smooth mappings from V to V , over the above ring of invariants, are given by bi in Table11 . Therefore, the normal form for the D4-equivariant map f in (1) is f(x) = 6 X i=1 Ai( 1(x); :::; 4(x))bi(x) (27) where, Ai are smooth functions.... ..."

### Table 1. Performance comparison of conformal geometric maps. Harmonic Maps Conformal Maps Least Squares Conformal Maps

2006

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### TABLE I PERFORMANCE COMPARISON OF QUASI-CONFORMAL MAPS. Harmonic Maps Conformal Maps Least Squares Conformal Maps

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### Table 1: Invariants and basic equivariants of D3 acting as #2 #3. The parameters have been chosen in such a way that they unfold the linear part. In the sense of singularity theory [8] one would like to show that all other equivariant functions satisfying f(0) = 0 and fx(0) = 0 are given by a nonlinear change of coordi- nates S(x; )f(X(x; ); ( )) preserving the equivariance (G-contact equivalence). Now we discuss the bifurcation points. The linearization along the trivial solution is given by

"... In PAGE 4: ... We use a REDUCE-program described in [6] to derive the general form of f. The rst result of the program is that the ring of invariant polynomials IR[x] (D3) is generated by four invariants 1(x); :::; 4(x) given in Table1 . The module of equivariant polynomial mappings IR[x]3 (D3) is generated over the ring IR[x] (D3) by some basic equivariants b1; : : :; b6 (see Table 1).... In PAGE 4: ... The rst result of the program is that the ring of invariant polynomials IR[x] (D3) is generated by four invariants 1(x); :::; 4(x) given in Table 1. The module of equivariant polynomial mappings IR[x]3 (D3) is generated over the ring IR[x] (D3) by some basic equivariants b1; : : :; b6 (see Table1 ). For the computation of fundamental invariants see also [12].... ..."

### TABLE II RECOGNITION RESULTS OF LEAST SQUARES CONFORMAL MAPS, SPHERICAL HARMONIC SHAPE CONTEXTS AND SURFACE CURVATURE TECHNIQUE.

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### Table 6: Timings for computation of equivariants for various variants.

### Table 2. Recognition results of conformal geometric maps and surface curvature technique. Recognition Result Harmonic Conformal Least Squares Surface

2006

"... In PAGE 7: ... In each experiment, we randomly select a single face from each subject for the gallery and use all the remaining faces as the probe set. The average recognition results from 20 exper- iments (with different randomly selected galleries) are re- ported in Table2 . In this experiment, conformal geometric maps perform 12.... ..."

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### Table 2. Deviation of harmonics from equal tempered scale #Harm max. deviation mean abs deviation

"... In PAGE 2: ... In this case, the deviation of harmonics from the equal tem- pered scale (the scale they are mapped to) has to be taken into account. Table2 shows mean and maximum deviation of the harmonic series from the closest equal tempered pitch frequency with respect to the number of harmonics. The fundamental is in tune with the scale.... ..."

### Table 2. Deviation of harmonics from equal tempered scale #Harm max. deviation mean abs deviation

"... In PAGE 2: ... In this case, the deviation of harmonics from the equal tem- pered scale (the scale they are mapped to) has to be taken into account. Table2 shows mean and maximum deviation of the harmonic series from the closest equal tempered pitch frequency with respect to the number of harmonics. The fundamental is in tune with the scale.... ..."